Quantum Nondemolition Measurements of Harmonic Oscillators

The complex amplitude $X_1+i{X}_2\equiv (x+\frac{\mathrm{ip}}{m\omega })e^{i\omega t}$ of a harmonic oscillator is constant in the absence of driving forces. Although the uncertainty principle forbids precise measurements of $X_1$ and $X_2$ simultaneously ($\Delta X_1\Delta X_2>~\frac{\hslash }{2m\omega }$), $X_1$ alone can be measured precisely and continuously ("quantum nondemolition measurement"). Examples are given of measuring systems that do this job. Such systems might play a crucial role in gravitational-wave detection and elsewhere.